Consider Fraunhofer (far-field) Diffraction from an arbitrary aperture
whose width and height are about the same.
Let A = the source strength per unit area. Then each infinitesimal area
element dS emits a spherical wave that will contribute an amount dE to the
field at P (X, Y, Z) on the screen

The distance from dS to P is
 
dE   A ei (t  kr ) dS
 r 
r  X 2  (Y  y ) 2  ( Z  z ) 2
which must be very large compared
to the size (a) of the aperture and
greater than a2/ in order to satisfy
conditions for Fraunhofer
diffraction. Therefore, as before, for
OP  , we can expect A/r  A/R
as before (i.e., the behavior is
approximated as that of a plane
wave far from the source).
R2  X 2  Y 2  Z 2
Fig. 10.19 A rectangular aperture.
At point P (X, Y), the complex field is
calculated as follows:
~  A i (t kR )
ikYy / R
ikZz / R
E
e
e
dy
e
dz


R
b / 2
a / 2
b/2
 
kbY
2R
a/2
and   
~ A Aei (t  kR )  sin    sin   

E




R
    
and the time  averaged irradiance is

~
I (Y , Z )  Re E

2
T
 sin   
 I (0)





2
 sin   


  
2
kaZ
2R
For x, y << R, we can approximate r as follows:

r  X 2  (Y  y ) 2  ( Z  z ) 2

1/ 2

 X 2  Y 2  2 yY  y 2  Z 2  2 zZ  z 2

1/ 2
 [ R 2  y 2  z 2  2(Yy  Zz )]1/ 2  R [1  ( y 2  z 2 ) / R 2  2(Yy  Zz ) / R 2 ]1/ 2


 R 1  Yy  Zz  / R 2  ...
Therefore,
~
E
 dE 
Aperture

Aperture
A
r
e i (t  kr ) ds 
A
R
e i (t  kR )
ik (Yy  Zz ) / R
e
ds

Aperture
Thus, for the specific geometry of the rectangular aperture:
~ 
E  A ei (t  kR )  eikYy / R dy  eikZz / R dz
R
b / 2
a / 2
b/2
Let
 
kbY
2R
a/2
and   
kaZ
2R
e i    e  i 
sin  
ikYy / R
Then  e
dy  b
b
and
2i 

b / 2
b/2
a/2
e
a / 2
ikZz / R
e i    e  i 
sin  
dz  a
a
2i 

Therefore, the resulting complex field at point P on the screen for a rectangular
aperture having area A = ab is given by
~ A Aei (t  kR )  sin    sin   

E




R
    

~
I (Y , Z )  Re E

2
T
 sin   
 I (0)





and the time  averaged irradiance is
2
 sin   


  
2
where I(0) is the intensity at the center of the screen at point P0 (Y = 0, Z = 0).
A typical far-field diffraction pattern is shown in Fig. 10.20. Note that when 
= 0 or  = 0, we get the familiar single slit pattern.
The approximate locations of the secondary maxima along the -axis (which
is the Y- axis when  = 0 or Z = 0) is given by m  3/2, 5/2, 7/2...
Since sin  = 1 at these maxima, the relative irradiances along the -axis are
approximated as
I (Y ,0) 
I (0)
 m 2
and similarly for the   or Z  axis, I (0, Z ) 
I (0)
 m 2
where m  3/2, 5/2, 7/2...  I  1  ~ 1  ~ 1  ~ 1
22
62
122
Square aperture in which a = b.
kbY
kaZ
 
and   
2R
2R
where m= m  3/2, 5/2, 7/2... are the
positions of the secondary maxima and
1
1
1
I 1 ~
 ~
 ~
22
62
122
Distribution of
irradiance, I(Y,Z)
 (Z)
 (Y)
Distribution of
electric field, E(Y,Z)
A very important aperture shape is the circular hole, as this involves the natural
symmetry for lenses. Such a symmetry suggests the need for cylindrical
coordinates:
z   cos  ,
y   sin  , Z  q cos , Y  q sin 
ds  dydz  dd
~
E
 Ae i (t  kR )
R
a
and the complex field is
2
 
i ( kq / R ) cos(   )

d

d

e

0 0
e
ik (Yy  Zz ) / R
We are
calculating the
field E on the
screen  as a
function of the
screen’s radial
coordinate q.
Note that the term eik (Yy  Zz ) / R involves :
yY  zZ  q(sin  sin   cos  cos )
 q cos(  )
Fig. 10.21 Circular
Aperture Geometry
The circular shape of the aperture results in complete axial symmetry.
Therefore, the solution must be independent of . Therefore, we are permitted
to set  = 0. We are therefore led to evaluate the integral:
2
1
i ( kq / R ) cos
d  J 0 (u ) 
0 e
2
2
iu cos v
e
dv,

with v  
and u  kq / R
0
where J0(u) is the Bessel function (BF) of order zero. More generally the BF
of order m, Jm(u), is represented by the following integral:
J m (u ) 
i
(  m ) 2
2
i ( mvu cos v )
e
dv,

0
Bessel functions are slowly
decreasing oscillatory
functions very common in
mathematical physics.
Fig. 10.22
Therefore the field is expressed as
~  Aei (t kR )
E
2  J 0 (kq / R) d
R
 0
a
Another important property of the Bessel
Function is the recurrence relation that
connects BFs of consecutive orders:


d m
u J m (u )  u m J m 1 (u )
du
d
when m  1,
u J1 (u )  u J 0 (u )
du
u
u
d
   J 0 ( )d  
 J1 ( ) d
d
0
0




  J1 ( )
Let
a

u
0
 u J1 (u )
  dummy var .
R
w  kq / R, then d    dw and
 kq 
2


J
k

q
/
R

d


(
R
/
kq
)
 0
 0

kaq / R
J
w0
0
( w) wdw
We can express the field, using the recurrence relation, as
~  Ae
E

i (t  kR )
R
a
2
 
i ( kq / R ) cos(   ) 

d

d

e

0 0
 Aei (t  kR )
R
 Aei (t kR )
R
2
a
 J
0
(kq / R) d
0
 R 
 J1 kaq / R 
2a 
 kaq 
2
Therefore, the irradiance at point P on the screen is
1 ~ ~*
  A   J kaq / R 
2
EE  2 A   1
;
A


a



2
R
kaq
/
R

 

2
I (q) 
2
It is useful to examine the series representation of the Bessel Functions:
n2k
(1) ( x / 2)
J n ( x)  
k 0 k! ( n  k  1)

k
where (n)  (n  1)!
3
x 1 x
 J1 ( x)      ...
2 22
Consider the limit near x = 0 : lim
x 0
J 1( x ) 1

x
2
Therefore, the irradiance at P0 when q = 0 is
AA  1 
1  AA
I (0)  2
    

2 R 
 R  2
2
2
2
and the Irradiance becomes
 2 J kaq / R  
q
I (q )  I (0)  1
;
and
with
sin





kaq
/
R
R


2
 2 J1 ka sin  
 I ( )  I (0) 

 ka sin  
(Fig. 10.21)
2
We usually express the irradiance as a function of the angular deviation 
from the central maximum at point P0. The large central maximum is called
the Airy Disk, which is surrounded by the first dark ring corresponding to
the first zero of J1(u). J1(u) = 0 when u = 3.83 or kaq1/R = kasin1= 3.83.
2 aq1
3.83 R
R
 3.83  q1 
 1.22
 R
 2a
2a
 sin 1  q1 / R  1.22 / D
kaq1 / R 
D = 2a is the diameter
of the circular hole
Note that ~84% of the light is contained in the Airy Disk (i.e. 0  kasin  3.83)
Airy Disk
Airy Disk
Airy Rings with different
hole diameters
D = 0.5 mm
Suppose that the aperture is a lens which
focuses light on a screen:
Entrance Pupil
(Aperture)
screen
Converging
lens
f
Airy
Disk
D = 1.0 mm
D
Since f  R  q1  1.22 f / D
which gives the radius of the Airy disk on
the screen.
Analysis of overlapping images
using Airy rings
Rays arriving from two
stars and striking a lens
Suppose that we image two equal irradiance point sources
(e.g., stars) through the objective lens of a telescope. The
angular half-width of each image point is q1/f = sin    .
If the angular separation of the stars is  and  >>  the
images of the stars will be distinct and well resolved.
Fig. 10.24 Overlapping Images
Rayleigh’s criterion for the minimum resolvable
angular separation or angular limit of resolution
Half-angle of an
airy disk:
1.22
1 
D
f
)l(min
Fig. 10.25
Rays from two stars
If the stars are sufficiently close in angle so that the center
of the Airy disk of star 1 falls on the first minimum (dark
ring) of the Airy pattern of star 2, we can say that the two
stars are just barely resolved. In this case, we have 1 =
( )min = q1/f  1.22  / D  (l)min 1.22 f/D . This is
Rayleigh’s criterion for angular or spatial resolution.
Another criterion for resolving two objects has
been proposed by C. Sparrow.
At the Rayleigh limit there is a central
minimum between adjacent peaks.
Further decrease in the distance between two
point sources will cause this minimum (dip) to
disappear such that
Sp
 d 2 I (r ) 
0

2 
dr

 r Sp
The resultant maximum will therefore have
a broad flat top when the distance between
the peaks is r = Sp, and serves as the
Sparrow criterion for resolving two point
objects.
Diffraction Gratings
Transmission Grating is made by
scratching rulings or notches onto a clear
flat plate of glass. Each notch serves as a
source of scattering that affects radiating
secondary sources, in much the same way
as for a multiple-slit diffraction array.
When the phase conditions are met
through OPD =  = m, constructive
interference is observed.
Oblique incidence (i > 0)
a sin  i  CD
a sin  m  AB
from the geometry.
Therefore AB  CD  a(sin  m  sin  i )  m
m = 0 (zeroth order), m = 1 (first order),
m = 2 (second order), m = 3 (third order)
The diffraction grating can also be constructed as a reflection
grating. The principals and conditions for constructive
interference are the same as that for a transmission grating.
AB  CD  a(sin m  sin i )  m
Most commercial gratings for spectroscopy are constructed
with a Blaze angle  to control the efficiency of diffraction
for a particular  and order m.
Controlling the irradiance distribution of
diffracted orders using a Blazed grating.
Consider the situation such that: i = 0 so
m = 0, 0 = 0. For specular reflection i r = 2 and so most of the diffracted
irradiance is concentrated near r = -2.
This will correspond to a particular nonzero order in which m = -2 and asin(-2)
= m.
Most of the incident light undergoes specular reflection, similar to a plane mirror,
and this occurs when i = m and m = 0 for the zeroth order beam. The problem is
that most of the irradiance is wasted for the purpose of spectroscopy. It is
possible to shift the reflected energy distribution into a higher order (m = 1) in
which m depends on . It is possible to change the distribution of the specular
reflection by changing the blaze angle  so that the first order diffraction is
optimized for a particular range of wavelengths.
Schematic from the Fluorescence Group, University
of California, Santa Barbara, USA
Monochromators with micrometer adjustable entrance and exit slit widths
• Exit slit determines spectral resolution of the instrument
• Resolution is determined by the product of the monochromator linear
dispersion (nm/mm) and the slit width
• Monochromator resolution depends on the grating pitch and the
(focal) length of the monochromator
• For PTI monochromator with 1200 groove/mm grating the reciprocal
linear dispersion is 4 nm/mm.
• 1 turn of the slit micrometer = 0.5 mm slit opening   = 2 nm
spectral resolution. Note that since E = h = hc/  E = (-hc/2) 
CL Intensity (a.u.)
P=200 W/cm
2
T =50 K (b)
GaN CR
SQW Sample
x 2.4
S1
x 1.04
S2
x1
S4
x 1.08
S8
320
360
400
 (nm)
440
480
Example of luminescence
spectra measured with a
grating monochromator on the
left for GaN/InGaN multiple
quantum well (QW) samples.
The maximum spectral
resolution is obtained for the
narrowest slit widths.
Sit widths narrow (top)
and wide (bottom)
Cathodoluminescence (CL) - Light emitted by the injection of high-energy electrons.
SEM Vacuum Sy stem
SEM Imaging
& Control
External
Y X
Condens er
Lens
Photomultiplier Tube (PMT)
or Ge p-i-n detector
Three-axis
Manipulator
Electron
Beam
Scan Coils
Detector
Ellipsoidal
Mirror
Coherent Optical
Fiber Bundle
Sample
Polarizer
Liquid He Stage
Monochromator
or Spectrograph
Controller
Lock-in Amp/Pho ton Counting Un it
Computer
CO M
ADC/
Counter
Imaging and DAC Y
Spectros copy DAC X
From Prof. Rich’s Laboratory for Optical Studies of Quantum Nanostructures
Schematic diagram of optical collection system and data acquisition
setup in our CL system.
Fresnel (Near-Field) Diffraction
The basic idea is to start again with the Huygen’s-Fresnel principle for
secondary spherical wave propagation. At any instant, every point on the
primary wavefront is envisioned as a continuous emitter of spherical secondary
wavelets. However, no reverse wave traveling back toward the source is
detected experimentally.
1
K ( )  1  cos  
2
Therefore, in order to introduce a
realistic radiation pattern of
secondary emitters, we introduce
the inclination factor, K() =
(1+cos)/2 which describes the
directionality of secondary
emissions. K(0) = 1 and K() = 0.
A rectangular aperture in the near-field (Fresnel Diffraction)
The monochromatic point source S and the point P on a screen are placed
sufficiently close to the aperture where far-field conditions are no longer
applicable. Consider a point A in the aperture whose coordinates are (y,z).
The location of the origin O is determined by a perpendicular line from
the source S to the aperture . The field
contributions at P from the secondary
sources on dS (area element at point A)
is given by
K ( ) A
cosk (   r )  t dS
r
K ( ) 0

cosk (   r )  t dS ,
r
dEP 
where 0 is the source strength at
S, A is the secondary wavelet
source strength per area, and
A = 0 is obtained from the
Huygen’s-Fresnel formalism.
In the case where the dimensions of the aperture are small compared to  and r,
we can assume primarily forward propagation in the secondary spherical waves
so that K()  1 and 1/r  1/0r0. Also, from the figure the geometry yields:
    y  z
2
0
2

2 1/ 2
and

r r y z
2
0
2

2 1/ 2
Expand both terms in a binomial series for small y and z:
 0  r0
  r   0  r0   y  z 
2  0 r0
2
2
Note that this approximation contains quadratic terms that appear in the phase
whereas the Fraunhofer approximation contains only linear terms. Thus, we
can expect a greater sensitivity in the phase of the cosine for this near-field
treatment. The complex field at point P on the screen is therefore:
 it y 2 z 2
~  e
EP  0
 0 r0
ik (   r )
e
dydz

y1 z1
1/ 2
 2 0  r0  
 2 0  r0  
Let u  y 
 ; v  z


r

r
0 0
0 0




Change of Variables Gives
1/ 2
~ u2
u2
v2
v2
2
2
2
2

E
~  0
EP  
e i k (  0  r0 ) t    e i u / 2 du  e i v / 2 dv  u  e i u / 2 du  e i v / 2 dv
2 u1
 2 0  r0 
 u1
v1
v1
0
~
Eu 
e i k (  0  r0 ) t 
This is the unobstructed disturbance at P.
 0  r0 
w


Let A( w)   cos w2 / 2 dw
w
and
0
w
Since
e
i w 2 / 2


C ( w)   sin w2 / 2 dw
0
dw  A( w)  iC ( w)
0
~
Eu
~
A(u )  iC (u ) uu12 A(v)  iC (v) vv12 ,
Therefore at P, E P 
2
where A(w) and C(w) are called Fresnel Integrals; note that both are odd
functions of w.
Very often, we work in the limit of incoming plane-waves striking the aperture.
For example, a laser beam could strike the aperture. In this limit we let the radius
from the source to the aperture 0  . This results in an immediate
simplification for the change of variables:
1/ 2
 2 0  r0  
 2 
 2 0  r0  


Then u  y 

y
;
v

z



 r 
  0 r0 
 0
  0 r0 
~ ~
2
2
E P* E P I 0
A(u2 )  A(u1 )  C (u2 )  C (u1 )
IP 

2
4


1/ 2
 A(v )  A(v )
2 ~
2
I ~
 0 B12 (u ) B12 (v)
4
2
1

2
 C (v2 )  C (v1 )
2
1/ 2
1/ 2
 2 

 z 
 r0 

u

2
~
~
where B12 (u )  A(u )  iC (u )  B (u )
u1

u2
u1
Cornu Spiral
Elegant geometrical description of the Fresnel Integrals (Fig. 10.50).
Fig. 10.50 The Cornu Spiral
for a graphical representation
of the Fresnel integrals.
C(w)
A(w)
w
A(w)
C(w)
A(w) C(w)
A(w)
C(w)
w
A(w)
C(w)
Let
~
B ( w)  A( w)  iC ( w)    w  
Arc length along the curve : dl 2  dA2  dC 2
2
2

 2

w

w
2
2
2

 dw  dl  dw
dl   cos
 sin
2
2 

From the Integrals :
Therefore, values of w in B(w) correspond to arc length on the Cornu spiral.
(-1 mm, 1 mm)
Consider a 2-mm square aperture hole:
(y1, z2)
We are given that  = 500 nm, r0 = 4 m,
plane wave approx. is valid. Find the
O
irradiance at a point P on the screen along
the axis x, directly behind the center of
(y1, z1)
the aperture.
(-1 mm, -1 mm)
1/ 2
1/ 2
 2 
2

   1103 m 
Then u1  y1 

7

r
5

10
m

4
m


 0
u2  1.0, v1  1.0, v2  1.0


z
(1 mm, 1 mm)
(y2, z2)
y
(y2, z1)
(1 mm, -1 mm)
r0
 1.0
P
For u1  1, u2  1
~
~
B (u2 )  0.7799  i 0.4383 and B (u1 )  0.7799  i 0.4383
~
~
 B 12 (u )  B 12 (v)  20.7799  i 0.4383

2 ~
2
I0 ~
I
2
2
Therefore I P 
B 12 (u ) B 12 (v)  0 (4 2 ) 0.7799  0.4383
4
4

2
 4(0.64) I 0  2.56 I 0  I 0
Notice that there is an increase of the irradiance at the center point P on the
screen by 256% relative to the unobstructed intensity due to a redistribution of
the energy.
z
(1.1 mm, 1 mm)
In order to find the irradiance 0.1 (-0.9 mm, 1 mm)
(y1, z2)
(y2, z2)
mm to the left of center, move the
aperture to the right relative to
y
the OP line. While y1 and y2 are
O
shifted, z1 and z2 remain
unchanged. Then we have u2 =
(y2, z1)
(y1, z1)
1.1, u1 = -0.9, v2 = 1.0, v1 = -1.0.
(1.1 mm, -1 mm)
(-0.9 mm, -1 mm)
r0
1/ 2
1/ 2
 2 
2


3


u1  y1 


0
.
9

10
m


7


r
5

10
m

4
m


 0
u2  1.1, v1  1.0, v2  1.0


 0.9
P
~
~
Then B (u1 )  B (0.9)  0.7648  i 0.3398
~
~
B (u2 )  B (1.1)  0.7638  i0.5365
~
 B12 (u )  1.5286  i0.8763
~
~
Also B12 (v) remains unchanged : B12 (v)  20.7799  i0.4383
Thus
I P 

2 ~
2
I0 ~
I
2
2
B 12 (u ) B 12 (v)  0 (4) 0.7799  0.4383
4
4
 1.5286  0.8763   2.485I
2
2
The decrease in the irradiance (2.485I0 < 2.56I0) for a small 0.1 mm shift to
the left (or right) of center on the screen shows that the center position is a
relative maximum (see Cornu spiral on the next slide).
Note that if the aperture is completely opened:
u1  v1  , u2  v2  
2
2
I0  2   2 
 2
  I0
I P   2


4  2   2 
which must equal to the
unobstructed intensity as a check.
0
1.1
C(w)
~
B12 (u )  1.762
~
B12 (u )
~
B12 (u )
A(w)
-0.9
~
B12 (u )  1.789
The decrease in the complex vector
length from the position of the central
peak shows that the central position is a
maximum.
We can apply this formalism for Fresnel diffraction by a long narrow slit in which
y1 , u1  , y2 , u2  
Therefore I P 
~
~
 B12 (u )  2ei / 4 , B12 (u )  2
2 ~
2
2
I0 ~
I ~
B 12 (u ) B 12 (v)  0 B 12 (v)
4
2
b = z1 – z2 = slit width and let v = v2 – v1 which is a string of length v
lying along the Cornu spiral (next slide).
Suppose that v = 2. At point P, opposite point O in the aperture, the
aperture and the screen are centered symmetrically and the string is centered
at point Os. If the aperture is moved up or down, the arc length of the string
remains constant, but the length of the vector B12(v) changes, as before.
~
B12 (v)
~
B12 (v)
It should be apparent that the length of
B12 (and the intensity at point P on the
screen) will oscillate as the string slides
around one of the spirals, which is
equivalent to the slit moving up or down
with respect to a reference point on the
screen, as shown in the previous slide.
~
B12 (3.5)
It is also possible to visualize a clear
minimum at the center of the near field
diffraction pattern on the screen by
considering the an arc-length of w = 3.5.
Any change in the slit position will give
and increase in B12 and therefore an
~
B12 (2.5) increase in irradiance.
It is apparent that the slit width has a
marked effect on whether the central
position is a maximum or local
minimum. Also note the oscillation
in irradiance for positions beyond
the width of the slit in both cases.